LECTURE 11

Introduction to Econometrics

Autocorrelation

November 29, 2016

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ON PREVIOUS LECTURES

I We discussed the specification of a regression equation

I Specification consists of choosing:

1. correct independent variables2. correct functional form3. correct form of the stochastic error term

I We talked about the choice of independent variables andtheir functional form

I We started to talk about the form of the error term - wediscussed heteroskedasticity

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ON TODAY’S LECTURE

I We will finish the discussion of the form of the error termby talking about autocorrelation (or serial correlation)

I We will learn

I what is the nature of the problem

I what are its consequences

I how it is diagnosed

I what are the remedies available

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NATURE OF AUTOCORRELATION

I Observations of the error term are correlated with eachother

Cov(εi, εj) 6= 0 , i 6= j

I Violation of one of the classical assumptions

I Can exist in any data in which the order of theobservations has some meaning - most frequently intime-series data

I Particular form of autocorrelation - AR(p) process:

εt = ρ1εt−1 + ρ2εt−2 + . . .+ ρpεt−p + ut

I ut is a classical (not autocorrelated) error termI ρk are autocorrelation coefficients (between -1 and 1)

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EXAMPLES OF PURE AUTOCORRELATION

I Distribution of the error term has autocorrelation nature

I First order autocorrelation

εt = ρ1εt−1 + ut

I positive serial correlation: ρ1 is positiveI negative serial correlation: ρ1 is negativeI no serial correlation: ρ1 is zeroI positive autocorrelation very common in time series dataI e.g.: a shock to GDP persists for more than one period

I Seasonal autocorrelation (in quarterly data)

εt = ρ4εt−4 + ut

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EXAMPLES OF IMPURE AUTOCORRELATIONI Autocorrelation caused by specification error in the

equation:I omitted variableI incorrect functional form

I How can misspecification cause autocorrelation in theerror term?

I Recall that the error term includes the omitted variables,nonlinearities, measurement error, and the classical errorterm.

I If we omit a serially correlated variable, it is included in theerror term, causing the autocorrelation problem.

I Impure autocorrelation can be corrected by better choice ofspecification (as opposed to pure autocorrelation).

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AUTOCORRELATION

X

Y

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CONSEQUENCES OF AUTOCORRELATION

1. Estimated coefficients (β̂) remain unbiased and consistent

2. Standard errors of coefficients (s.e.(β̂)) are biased(inference is incorrect)

I serially correlated error term causes the dependent variableto fluctuate in a way that the OLS estimation procedureattributes to the independent variable

I Serial correlation typically makes OLS underestimate thestandard errors of coefficients

I therefore we find t scores that are incorrectly too high

⇒ The same consequences as for the heteroskedasticity

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DURBIN-WATSON TEST FOR AUTOCORRELATION

I Used to determine if there is a first-order serial correlationby examining the residuals of the equation

I Assumptions (criteria for using this test):

I The regression includes the intercept

I If autocorrelation is present, it is of AR(1) type:

εt = ρεt−1 + ut

I The regression does not include a lagged dependentvariable

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DURBIN-WATSON TEST FOR AUTOCORRELATION

I Durbin-Watson d statistic (for T observations):

d =

T∑t=2

(et − et−1)2

T∑t=1

e2t

≈ 2(1− ρ̂)

where ρ̂ is the autocorrelation coefficient

I Values:

1. Extreme positive serial correlation: d ≈ 02. Extreme negative serial correlation: d ≈ 43. No serial correlation: d ≈ 2

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USING THE DURBIN-WATSON TEST

1. Estimate the equation by OLS, save the residuals

2. Calculate the d statistic

3. Determine the sample size T and the number ofexplanatory variables (excluding the intercept!) k′

4. Find the upper critical value dU and the lower criticalvalue dL for T and k′ in statistical tables

5. Evaluate the test as one-sided or two-sided (see next slides)

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ONE-SIDED DURBIN-WATSON TEST

I For cases when we consider only positive serial correlationas an option

I Hypothesis:

H0 : ρ ≤ 0 (no positive serial correlation)HA : ρ > 0 (positive serial correlation)

I Decision rule:I if d < dL reject H0

I if d > dU do not reject H0

I if dL ≤ d ≤ dU inconclusive

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DURBIN-WATSON CRITICAL VALUES FOR ONE-SIDED

TEST

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TWO-SIDED DURBIN-WATSON TEST

I For cases when we consider both signs of serial correlation

I Hypothesis:

H0 : ρ = 0 (no serial correlation)HA : ρ 6= 0 (serial correlation)

I Decision rule:

I if d < dL reject H0

I if d > 4− dL reject H0

I if d > dU do not reject H0

I if d < 4− dU do not reject H0

I otherwise inconclusive

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DURBIN-WATSON CRITICAL VALUES FOR TWO-SIDED

TEST

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EXAMPLE

I Estimating housing prices in the UK

I Quarterly time series data on prices of a representativehouse in UK (in £)

I Explanatory variable: GDP (in billions of £)

I Time span: 1975 Q1 - 2011 Q2

I All series are seasonally adjusted and in real prices (i.e.adjusted for inflation)

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EXAMPLE

60000

80000

100000

120000

140000

160000

180000

200000

220000

1975 1980 1985 1990 1995 2000 2005 2010

Pri

ce o

f re

pre

senta

tive h

ouse

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EXAMPLE

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EXAMPLE

I We test for positive serial correlation:

H0 : ρ ≤ 0 (no positive serial correlation)HA : ρ > 0 (positive serial correlation)

I One-sided DW critical values at 95% confidence forT = 146 and k′ = 1 are:

dL = 1.72 and dU = 1.74

I Decision rule:I if d < 1.72 reject H0

I if d > 1.74 do not reject H0

I if 1.72 ≤ d ≤ 1.74 inconclusive

I Since d = 0.02 < 1.72, we reject the null hypothesis of nopositive serial correlation

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ALTERNATIVE APPROACH TO AUTOCORRELATION

TESTING

I Suppose we suspect the stochastic error term to be AR(p)

εt = ρ1εt−1 + ρ2εt−2 + . . .+ ρpεt−p + ut

I Since OLS is consistent even under autocorrelation, theresiduals are consistent estimates of the stochastic errorterm

I Hence, it is sufficient to:

1. Estimate the original model by OLS, save the residuals et

2. Regress et = ρ1et−1 + ρ2et−2 + . . .+ ρpet−p + ut

3. Test if ρ1 = ρ2 = . . . = ρp = 0 using the standard F-test

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BACK TO EXAMPLE

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BACK TO EXAMPLE

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REMEDY: WHITE ROBUST STANDARD ERRORS

I Note that autocorrelation does not lead to inconsistentestimates, only to incorrect inference - similar toheteroskedasticity problem

I We can keep the estimated coefficients, and only adjust thestandard errors

I The White robust standard errors solve not onlyheteroskedasticity, but also serial correlation

I Note also that all derived results hold if the assumptionCov(x, ε) = 0 is not violated

I First make sure the specification of the model is correct,only then try to correct for the form of an error term!

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SUMMARY

I Autocorrelation does not lead to inconsistent estimates,but it makes the inference wrong (estimated coefficientsare correct, but their standard errors are not)

I It can be diagnosed using

I Durbin-Watson testI Analysis of residuals

I It can be remedied by

I White robust standard errors

I Readings:I Studenmund, Chapter 9I Wooldridge, Chapter 12

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